No because you would be in check.
Wouldn't it make sense to be able to take with the king?

No. Pinned pieces still can deliver check and the rule is that a move can't be made if that will result in the king being in check.
Think about it this way. If you could move into check then so could your opponent. However, your opponent's move would allow your king to be captured which would end the game; you wouldn't be able to make your move.
Hi,
So, in the situation below... wouldn't it be normal for me to be able to take with the rook with my king? Since the other rook is pinned, it wouldn't be able to move, right? He would have to make a king move and then I could move my king away from danger, right? What gives, chess rules?